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12 days ago

. Andrew made an error in determining the polynomial equation of smallest degree whose roots are 3, 2+2i

Mathematics

1 answer:

PIT_PIT [3.9K]12 days ago

7 0

Answer:

Error made by Andrew: He identified incorrect factors based on the roots.

Step-by-step explanation:

The roots of the polynomial consist of: 3, 2 + 2i, 2 - 2i. By the factor theorem, if a is a root of the polynomial P(x), then (x - a) must be a factor of P(x). According to this premise:

(x - 3), (x - (2 + 2i)), (x - (2 - 2i)) represent the factors of the polynomial.

<pBy simplification, we obtain:

(x - 3), (x - 2 - 2i), (x - 2 + 2i) as the respective factors.

This is where Andrew's mistake occurred. Factors should always be in the form (x - a), not (x + a). Andrew expressed the complex factors incorrectly, resulting in an erroneous conclusion.

Thus, the polynomial can be expressed as:

$(x - 3)(x - 2 - 2i)(x - 2+2i)=0\\\\ (x-3)(x^{2}-2x+2xi-2x+4-4i-2xi+4i-4i^{2})=0\\\\ (x-3)(x^{2}-4x+4+4)=0\\\\ (x-3)(x^{2}-4x+8)=0\\\\ x^{3}-4x^{2}+8x-3x^{2}+12x-24=0\\\\ x^{3}-7x^{2}+20x-24=0$

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Which of the following functions is graphed below?

Svet_ta [4321]

Answer:

Option (B)

Step-by-step explanation:

The given question is incomplete; please refer to the attached document for the full details.

In the attached graph,

The parent function is defined as an absolute value function,

f(x) = |x|

When this graph is moved four units to the left, the translation rule becomes,

f(x) → f(x + 4)

Consequently, the new function following this translation is,

g(x) = f(x + 4) = |x + 4|

Now, with the graph shifted down by two units, the translated function becomes,

h(x) = g(x) - 2

h(x) = |x + 4| - 2

When reformulated as an equation, the graph can be represented by

⇒ y = |x + 4| - 2

Thus, Option (B) is the correct answer.

8 0

14 days ago

After a 5% pay raise , Hermione is earning $21,000 per year. What was she earning before the pay raise?

zzz [4022]

Answer:

$\$20,000$

Explanation step-by-step:

Let

x represent Hermione's salary prior to the raise

Given:

$100\%+5\%=105\%=105/100=1.05$

Therefore, the equation that expresses this relationship is

$1.05x=21,000$

To find x, divide both sides by 1.05:

$x=21,000/1.05$

$x=\$20,000$

7 0

13 days ago

If y varies directly as x, and y is 400 when x is r and y is r when x is 4, what is the numeric constant of variation in this

Leona [4166]

Answer:

Step-by-step explanation:

If there’s a direct correlation between variables x and y, it can be expressed as

y = kx,

where y is the dependent variable

x is the independent variable

k signifies the constant of variation

_____________________________

For the first scenario

y = 400

x = r

using y = kx, we can derive the relation:

400 = kr

To find k here:

k = 400/r

For the second scenario

y = r

x = 4

again using y = kx, the relationship is given by:

r = 4k

Solving for k:

k = r/4

Since both conditions provide the same equation, k will have the same value in both cases.

Thus, we conclude:

400/r = r/4

=> 400*4 = r*r

=> 1600 = r^2

$\sqrt{r^{2} } = \sqrt{1600} \\r = 40$

Hence, r is calculated to be 40.

Thus, k = r/4 = 40/4 = 10.

The constant of variation is confirmed to be 10, which is the correct answer.

Verification:

k = 400/r = 400/40 = 10

7 0

8 days ago

Read 2 more answers

Billy graphed the system of linear equations to find an approximate solution. y = A system of equations. y equals negative Start

babunello [3635]

Response:

Second option: $(2.2, -1.4)$

Third option: $(2.2, -1.35)$

Detailed explanation:

The missing graph has been provided.

The attached image illustrates the graphing of the following system of linear equations:

$\left \{ {{y=-\frac{7}{4}x+\frac{5}{2}} \atop {y=\frac{3}{4}x-3}} \right.$

Notice the intersection of the lines.

According to the definition, if lines in a system of equations intersect, then there is only one solution. This implies that the intersection point is the solution to that system. This can be expressed as:

$(x,y)$